4. Let M-(E, F) be a matroid and let E'-{e e E : {e) e F}. Show that dim(P)-el (where E' is the s...
Exercise 25: Let f: [0,1R be defined by x=0 fx)/n, m/n, with m, n E N and n is the minimal n such that z m/n x- m/n, with m,n E N and n is the minimal n such that x a) Show that L(f, P) = 0 for all partitions P of [0, 1]. b) Let m E N. Show that the cardinality of the set A :-{х є [0, 1] : f(x) > 1/m} is bounded by m(m...
(e) Let V and W be finite dimensional vector spaces, dim V-n and din W-m. Show that there is a bijection between the set Bil V, W;R) of all bilinear functions on V and W and the set GLn(m)R of all matrices of order n x m Hint: Uselal & rbl (e) Let V and W be finite dimensional vector spaces, dim V-n and din W-m. Show that there is a bijection between the set Bil V, W;R) of all...
(10) Let ū ER. Show that M = {ū= | ER*:ūū= 0) is a subspace of R'. Definition: (Modified from our book from page 204.) Let V be a subspace of R". Then the set of vectors (61, 72, ..., 5x} is a basis for V if the following two conditions hold. (a) span{61, 62,...,x} = V (b) {61, 62, ..., 5x} is linearly independent. Definition: Standard Basis for R" The the set of vectors {ēi, 72, ..., en) is...
a set function, λ on S by λ((a, b) F(b)--F(a) and λ(0) 1. Show that if Eİ, E2 E S then Ei n E2 ES and Ei ~ E2 is a finite disjoint union of 0. sets in S 2. Show that the o-algebra generated by S is the Borel o-algebra on R. 3. Show that if E and Ea are disjoint sets in S and A U S, then (A) A(E)+A(B2). 4, Show that if E. .. ova natn...
2. (a) Let B = {f1, f2, f3} be a subset of P2 where fi(x) = x² – 3, f2(x) = x2 – 2x and f3(x) = x. Show that B is a basis of P2. (b) Determine whether or not the following sets are subspaces of F. (i) X = {f € F | f(x) = a(x + cos x), a € R}. (ii) Y = {f EF | f(x) = ax + sin x, a € R}. (c)...
Let TRm → Rn be a linear transformation, and let p be a vector and S a set in R Show that the image of p + S under T is the translated set T(p) + T(S) n R What would be the first step in translating p+ S? OA. Rewrite p+ S so that it does not use sets. O B. Rewrite p+S so that it does not use vectors O c. Rewrite p + S as a difference...
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
Materials: ------------------------------------------------------------------ 9. Let f E (R" where R" is the standard Euclidean space (vector space Rn equipped with the Euclidean scalar product) (i) Explain why there are constants ai,....an R such that 21 ii) Obtain u R" such that f(x)-(1,2), х є R". (ii Explain why the correspondence f u establishedin) is 1-1, onto, and linear so that (R" and R" may be viewed identical. With the usual addition and multiplication, the sets of rational numbers, real numbers, and...
Problem 2 (Eigenvalues and Eigenvectors). (a) If R2 4 R2 be defined by f(x,y) (y,x), then find all the eigenvalues and eigenvectors of f Hint: Use the matrix representation. (b) Let U be a vector subspace (U o, V) of a finite dimensional vector space V. Show that there exists a linear transformation V V such that U is not an invariant subspace of f Hence, or otherwise, show that: a vector subspace U-0 or U = V, if and...
Problem 25 please -Sesin(2x)-9ecos(2x). 21. W = Span(B), where Br(x2e-4x , xe®, e-4x); f(x)--5x2r" + 2e-4-1e 22. W= Span(B),where B= ({x25, x5*, 5x)); f(x)--4x2 5x+9s5x-2(5x). 3 W Span(B), where B (Exsin(2x), xcos(2x), sin(2x), cos(2x)y): f(x) = 4x sin(2x) + 9x cos(20-5 sin(2x) + 8 cos(2x). 24, In Exercise 21 of Section 3.6, we constructed the matrix [D, of the derivative operator D on W- Span(B), where B e sin(bx), e" cos(bx)): Dls a a. Find [D 1g and [D'lg: Observe...